Hybridization between complementary nucleic acids is an implicit feature in the Watson-Crick model for DNA structure that is exploited for many applications of the biological and biomedical arts. For example, virtually all methods for replicating and/or amplifying nucleic acid molecules are initiated by a step in which a complementary oligonucleotide (typically referred to as a “primer”) hybridizes to some portion of a “target” nucleic acid molecule. A polymerase then synthesizes a complementary nucleic acid from the primer, using the target nucleic acid as a “template.” See, Kleppe et al., J. Mol. Biol. 1971, 56:341-361.
One particular application, known as the polymerase chain reaction, PCR, is widely used in a variety of biological and medical arts. For a description, see Saiki et al., Science 1985, 230:1350-1354. In PCR, two or more primers are used that hybridize to separate regions of a target nucleic acid and its complementary sequence. The sample is then subjected to multiple cycles of heating and cooling, repeatedly hybridizing and dissociating the complementary strands so that multiple replications of the target nucleic acid and its complement are performed. As a result, even very small initial quantities of a target nucleic acid may be enormously increased, or “amplified,” for subsequent uses (e.g., for detection, sequencing, etc.).
Multiplex PCR is a particular version of PCR in which several different primers are used to amplify and detect a plurality of different nucleic acids in a sample—usually ten to a hundred different target nucleic acids. Thus, the technique allows a user to amplify and evaluate large numbers of different nucleic acids simultaneously in a single sample. The enormous benefits of high throughput, speed and efficiency offered by this technique has made multiplex PCR increasingly popular. However, achievement of successful multiplex PCR usually involves empirical testing as existing computer programs that pick and/or design PCR primers have errors. In multiplex PCR, the errors become additive and therefore good results are seldom achieved without a substantialsome amount of trial and error. See, Markouatos et al., J. Clin. Lab Anal. 2002, 16(1):47-51; Henegarin et al., Biotechniques 1997, 23(3):504-11.
Other techniques that are widely used in the biological and medical arts exploit nucleic acid hybridization to detect target nucleic acid sequences in a sample. See, for example, Southern, J. Mol. Biol. 1975, 98:503-517; Denhardt, Biochem. Biophys. Res. Commun. 1966, 23:641-646; Meinhoth & Wahl, Anal. Biochem. 1984, 138:267-284. For instance, Southern blotting and similar techniques have long been used in which nucleic acid molecules from a sample are immobilized onto a solid surface or support (e.g., a membrane support). A target nucleic acid molecule of interest may then be detected by contacting one or more complementary nucleic acids (often referred to as nucleic acid “probes”) and detecting their hybridization to nucleic acid molecules on the surface or support. A signal generated by some detectable label on the probes is proportional to the amount of hybridization to the target.
Similar techniques are also known in which one or more nucleic acid probes are immobilized onto a solid surface or support, and a sample of nucleic acid molecules is hybridized thereto. Nucleic acid arrays, for example, are known and have become increasingly popular in the art. See, e.g., DeRisi et al., Science 1997, 278:680-686; Schena et al., Science 1995, 270:467-470; and Lockhart et al., Nature Biotech. 1996, 14:1675. See also, U.S. Pat. No. 5,510,270 issued Apr. 23, 1996 to Fodor et al. Nucleic acid arrays typically comprise a plurality (often many hundreds or even thousands) of different probes, each immobilized at a defined location on the surface or support. A sample of nucleic acids (for example, an mRNA sample, or a sample of cDNA or cRNA derived therefrom), that may be detectably labeled, may then be hybridized to the array. Hybridization of those nucleic acids to the different probes may be assessed, e.g., by detecting labeled nucleic acids at each probe's location on the array. Thus, hybridization techniques using nucleic acid arrays have the potential for simultaneously detecting a large number of different nucleic acid molecules in a sample, by simultaneously detecting their hybridization to the different probes of the array.
The successful implementation of all techniques involving nucleic acid hybridization (including the exemplary techniques described, supra) is dependent upon the use of nucleic acid probes and primers that specifically hybridize with complementary nucleic acids of interest while, at the same time, avoiding non-specific hybridization with other nucleic acid molecules that may be present. For a review, see Wetmur, Critical Reviews in Biochemistry and Molecular Biology 1991, 26:227-259. These properties are even more critical in techniques, such as multiplex PCR and microarray hybridization, where a plurality of different probes or primers is used, each of which may be specific for a different target nucleic acid.
Duplex stability between complementary nucleic acid molecules is frequently expressed by the duplex's “melting temperature”, Tm. Roughly speaking, the Tm indicates the temperature at which a duplex nucleic acid dissociates into single-stranded nucleic acids. Nucleic acid hybridization may be performed at a temperature just slightly below the Tm, so that hybridization between a probe or primer and its target nucleic acid is optimized, while minimizing non-specific hybridization of the probe or primer to other, non-target nucleic acids. Duplex stability and Tm are also important in applications, such as PCR, where thermocycling may be involved. During such thermocycling melting steps, it is important that the sample temperature be raised sufficiently above the Tm so that duplexes of the target nucleic acid and its complement are dissociated. In subsequent steps of reannealing, however, the temperature must be brought sufficiently below the Tm that duplexes of the target nucleic acid and primer are able to form, while still remaining high enough to avoid non-specific hybridization events. For a general discussion, see Rychlik et al., Nucleic Acids Research 1990, 18:6409-6412.
Traditionally, theoretical or empirical models that relate duplex stability to nucleotide sequence have been used to predict or estimate melting temperatures for particular nucleic acids. For example, Breslauer et al. (Proc. Natl. Acad. Sci. U.S.A. 1986, 83:3746-3750) describe a model for predicting melting temperatures that is widely used in the art, known as the “nearest neighbor model.” See also, SantaLucia et al., Biophys. Biomol. Struct. 2004, 33:415-440; Owczarzy et al., Biopolymers 1997, 44:217-239; and SantaLucia, Proc. Natl. Acad. Sci. USA. 1998, 95:1460-1465. Such models are usually calibrated or optimized for particular salt conditions, typically 1 M Na+. However, applications that exploit nucleic acid hybridization may be implemented in a variety of different salt conditions, including, for example, magnesium and potassium, with cation concentrations typically being on the order of magnitude of 0.001-1 M. Thus, melting temperatures for particular probes or primers in an assay are typically predicted by predicting a melting temperature at a first salt concentration using the nearest neighbor or other models, and then using another theoretical or empirical model to predict what effect(s) the salt conditions of the particular assay will have on that melting temperature.
Most existing models used to estimate Tm do so in solutions of some specific cation concentrations and then correct for presence and concentrations of all cations. Schildkraut et al. (Biopolymers 1965, 3:195-208) proposed the following formula to estimate nucleic acid melting temperatures at different sodium ion concentrations, [Na+]:Tm([Na+])=Tm(1M Na+)+16.6×log [Na+]  (Equation 1)where Tm(1M Na+) is the melting temperature of the DNA duplex in solution of 1 M sodium ions. Equation 1, above, is based on empirical data from the specific study of Escherichia coli genomic DNA in buffer of between 0.01-0.2 M [Na+]. Nevertheless, the use of this equation has been routinely generalized to model any DNA duplex oligomer pair. See, for example, Rychlik et al., Nucleic Acids Res. 1990, 18:6409-6412, Ivanov & AbouHaidar, Analytical Biochemistry 1995, 232:249-251; Wetmur, Critical Review in Biochemistry and Molecular Biology 1991, 26:227-259.
SantaLucia and Peyret analyzed data of 26 oligonucleotide duplexes and published correction equations for effects of sodium ions. They assumed that sodium ions change the transition entropy of duplex melting, but do not effect a value of ΔH0 (see SantaLucia, Proc. Natl. Acad. Sci. USA. 1998, 95:1460-1465 and Peyret, Ph.D. Thesis, Wayne State University, Detroit, Mich., pp. 128, section 5.4.2 (2000)), and derived the following equation,
                              1                                    T              m                        ⁡                          (                              Na                +                            )                                      =                              1                                          T                m                            ⁡                              (                                  1                  ⁢                  M                  ⁢                                                                          ⁢                                      Na                    +                                                  )                                              +                                                    0.368                ⁢                N                                            Δ                ⁢                                                                  ⁢                                  H                  0                                                      ×                          ln              ⁡                              [                                  Na                  +                                ]                                                                        (                  Equation          ⁢                                          ⁢          2                )            ΔH0 is the standard transition enthalpy predicted from a nearest-neighbor model and N is the number of phosphate groups in the duplex divided by 2. That is, N is typically for synthetic oligomers equal to number of base pairs decreased by one.
Also, U.S. Pat. No. 6,889,143 (incorporated herein by reference in its entirety) describes equations developed for varying sodium cation concentrations, taking into account the G-C content of the oligonucleotides,
                              1                                    T              m                        ⁡                          (                              Na                +                            )                                      =                              1                                          T                m                            ⁡                              (                                  1                  ⁢                  M                  ⁢                                                                          ⁢                                      Na                    +                                                  )                                              +                                    (                                                4.29                  ·                                      f                    GC                                                  -                3.95                            )                        ·                          10                              -                5                                      ·                          ln              ⁡                              [                                  Mon                  +                                ]                                              +                      9.40            ·                          10                              -                6                                      ·                                          (                                  ln                  ⁡                                      [                                          Mon                      +                                        ]                                                  )                            2                                                          (                  Equation          ⁢                                          ⁢          3                )            
While several equations were published to model relationships between monovalent cations (e.g., sodium) and DNA melting temperature (see, e.g., Owczarzy et al., Biochemistry 2004, 43:3537-3554), little is known about the effect of divalent cations. Corrections were previously suggested to explain effects of magnesium ions on DNA melting temperatures that are based on the assumption that stabilizing effects of magnesium ions are very similar to stabilizing effects of sodium ions and therefore Tm salt correction for sodium ions can be applied to solutions of magnesium ions using a simple adjustment. These corrections (Equations 6, 7, and 8, below) use Equation 4 where the square root of Mg2+ concentration is added to monovalent cation concentrations [Mon+] (e.g., Na+, Tris+, or K+) and the “equivalent effect” sodium concentration, [Na+]eq, is calculated,
                                          [                          Na              +                        ]                    eq                =                              β            ×                                          [                                  Mg                                      2                    +                                                  ]                                              +                      [                          Mon              +                        ]                                              (                  Equation          ⁢                                          ⁢          4                )            The monovalent cation concentration, [Mon+], is a sum of the concentrations of all monovalent cations in solution. In the pH range typically employed the H+ concentration is less than 10−5 M and need not be considered; however, H+ ions are not considered. For a typical PCR buffer, concentrations of K+ and Tris+ ions are summed,[Mon+]=[K+]+[Tris+]  (Equation 5)Values of the conversion factor β from 3.3 to 4 were suggested in published literature. The equivalent sodium concentration from Equation 4, [Na+]eq, may be combined with the Tm sodium correction equations 1 and 2. Three such correction equations were reported in the published literature,
                                          T            m                    ⁡                      (                          M              ⁢                                                          ⁢                              g                                  2                  +                                                      )                          =                                            T              m                        ⁡                          (                              1                ⁢                                                                  ⁢                M                ⁢                                                                                          ⁢                                                                                        ⁢                                  Na                  +                                            )                                +                      16.6            ×                          log              ⁡                              (                                                                                                                              4                          ·                                                                                    [                                                              Mg                                                                  2                                  +                                                                                            ]                                                                                                      +                                                                                                                                                [                                                  Mon                          +                                                ]                                                                                            )                                                                        (                  Equation          ⁢                                          ⁢          6                )            (Mitsuhashi, J. Clin. Lab. Analysis, 1996, 10:277-284)
                              1                                    T              m                        ⁡                          (                              Mg                                  2                  +                                            )                                      =                              1                                          T                m                            ⁡                              (                                  1                  ⁢                                                                          ⁢                  M                  ⁢                                                                          ⁢                                      NA                    +                                                  )                                              +                                                    0.368                ⁢                                                                  ⁢                N                                            Δ                ⁢                                                                  ⁢                                  H                  0                                                      ×                          ln              ⁡                              (                                  3.79                  ·                                                                                    [                                                  Mg                                                      2                            +                                                                          ]                                            +                                              [                                                  Mon                          +                                                ]                                                                                            )                                                                        (                  Equation          ⁢                                          ⁢          7                )            (von Ahsen et al., Clin. Chem. 2001, 47:1956-1961)
                              1                                    T              m                        ⁡                          (                              Mg                                  2                  +                                            )                                      =                              1                                          T                m                            ⁡                              (                                  1                  ⁢                                                                          ⁢                  M                  ⁢                                                                          ⁢                                      Na                    +                                                  )                                              +                                                    0.368                ⁢                                                                  ⁢                N                                            ΔH                0                                      ×                          ln              ⁡                              (                                                      3.3                    ·                                                                  [                                                  Mg                                                      2                            +                                                                          ]                                                                              +                                      [                                          Mon                      +                                        ]                                                  )                                                                        (                  Equation          ⁢                                          ⁢          8                )            (Peyret, Ph.D. Thesis, Wayne State University, Detroit, Mich., pp. 128, section 5.4.2 (2000)). In some of the above cases, the Tm correction function is expressed directly in terms of Tm (Equation 6), and in Equation 7 and 8 the Tm correction function is related to the reciprocal of Tm (1/Tm).
These equations were used to determine the Tm salt correction for a solution containing magnesium ions in the absence or presence of other monovalent ions.
Recently, Tan and Chen (Biophys. J 2006, 90:1175-1190) developed the “Tightly Bound Ion model” and proposed a new formula for dependence of melting temperatures on magnesium concentrations,
                              1                                    T              m                        ⁡                          (                              Mg                                  2                  +                                            )                                      =                              1                                          T                m                            ⁡                              (                                  1                  ⁢                                                                          ⁢                  M                  ⁢                                                                          ⁢                                      Na                    +                                                  )                                              -                                    0.00322              ×              Δ              ⁢                                                          ⁢                              g                el                            ×                              (                                                      N                    bp                                    -                  1                                )                                                    Δ              ⁢                                                          ⁢                              H                0                                                                        (                  Equation          ⁢                                          ⁢          9                )            where Δgel is the electrostatic free energy per base stack (kcal/mol),
                              Δ          ⁢                                          ⁢                      g            el                          =                                            (                                                                                          0.02                      +                                                                                                                                  1.18                                              N                        bp                        2                                                                                                        )                        ⁢                          ln              ⁡                              [                                  Mg                                      2                    +                                                  ]                                              +                                    (                                                                                          0.0068                      +                                                                                                                                  0.344                                              N                        bp                        2                                                                                                        )                        ⁢                                          (                                  ln                  ⁡                                      [                                          Mg                                              2                        +                                                              ]                                                  )                            2                                                          (                  Equation          ⁢                                          ⁢          10                )            The Equations 9 and 10 were proposed to be appropriate for duplexes with six or more base pairs in solutions where magnesium ions have dominant effects. These magnesium correction equations do not apply to mixed buffers where monovalent ions compete with magnesium ions.
Further studies on the correction of melting temperature include Nakano et al., Nucleic Acids Research 1999, 27:2957-2965; Williams et al., Biochemistry 1989, 28:4283-4291; Record, Biopolymers 1975, 14:2137-2158.
Notably, none of the Tm correction equations in the prior art consider the sequence of the polynucleotide or its G/C content value, fGC.
As will be demonstrated below, the above equations do not adequately predict melting temperatures in the presence of divalent cations. The errors are significant, in some cases as large as 15 C, and can adversely affect the performance of probes and primers in experiments and assays. The effects on melting temperature due to divalent cations, in the presence and/or absence of monovalent ions, differ significantly from the effects of sodium ions and are not adequately described in the equations above. Therefore, there is a significant need for methods of estimating and predicting melting temperatures with improved accuracy, especially for oligonucleotides in the presence of divalent cations. There further exists a need for methods of designing experiments in which the melting temperature of each oligonucleotide in the presence of divalent cations is optimized for the particular method or assay, such as PCR or other assay that involves nucleic acid hybridization. The present invention meets these needs by providing methods to more accurately predict the melting temperature of nucleic acids in buffers with divalent cations.
The citation or discussion of any reference in this section or elsewhere in the specification is made only to clarify the description of the present invention and is not an admission that any such reference is “prior art” against any invention described herein.